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The Yablo Paradox: An Essay on Circularity

Let us say that a sentence is periphrastic if and only if there is a single word in that sentence such that we can remove the word and the result (i) is grammatical, and (ii) has the same truth value as the original sentence. For example:

[1] Roy murdered someone.

is periphrastic, since it is equivalent to:

[2] Roy murdered.

Thus, a sentence is not periphrastic if, for any word in the sentence, the result of removing the word is not grammatical, or the result of removing that word has a different truth value.

It should be noted that I am introducing “periphrastic” as a technical term here, and its use as defined above is different from (but connected to) its meaning in everyday English (further, the meaning here is significantly different from its technical meaning in grammar).

The notion of a sentence being periphrastic in this sense is simple, and at first glance we might think that we will always be able to determine whether a sentence is periphrastic merely by checking all of the sentences that can be obtained by removing one word. But like many other simple notions such as truth and knowability, it leads to puzzles – in this case, puzzles very similar to the truth-teller:

This sentence is true.

Consider the following sentence:

[3] I am periphrastic.

(We assume here that “I” is an informal way for a sentence to refer to itself).

Purse Printing Portable Green Grape Fruit Leaves Small Bag Deluxe Receiving Now, [3] is either periphrastic or not. But there seems to be no way to determine which it is.

If [3] is periphrastic, then there must be some word that we can remove from [3] such that the result is grammatical and true (since if [3] is periphrastic then, since that is what it says, it is true). The only way to remove a single word from [3] and obtain a grammatical result is:

[4] I am.

This sentence states that [4] exists, and is clearly true. Thus, the claim that [3] is periphrastic (and hence true) is completely consistent.

If [3] is not periphrastic, however, then the result of removing any word must either be ungrammatical or true (since [3] says that it is periphrastic, and thus in this case [3] is false). But, again, the only way to remove a single word from [3] and obtain a grammatical result is again [4], which is true. Thus, the claim that [3] is not periphrastic (and hence false) is completely consistent.

There would seem to be no other evidence that could settle the matter. Thus, even though it is obvious that [3] is either periphrastic or it is not, determining which seems impossible.

Interestingly, although most notions that allow for the construction of a truth-teller type puzzle also admit of a Liar-like paradoxical construction, I have failed to find an example of a paradoxical sentence that involves the idea of sentences being periphrastic. The obvious candidate to look at is:

[5] I am not periphrastic.

But there is nothing paradoxical about [5] – it is perfectly consistent to assume that [5] is false, and hence (contrary to what it says) periphrastic, since:

[6] I am not.

Leaves Small Printing Bag Purse Green Fruit Receiving Grape Portable Deluxe is a sentence obtained from [5] via the deletion of a single word that then has the same truth value as [5] – they are both false.

Perhaps my readers can do better. Is there a clearly paradoxical sentence that can be constructed using the notion of a sentence being periphrastic?

Featured image: Pieces Of The Puzzle by Hans. Public domain via Pixabay.

Roy T Cook is Professor of Philosophy at the University of Minnesota – Twin Cities, a research fellow of the Minnesota Center for Philosophy of Science, and an associate fellow of the Northern Institute of Philosophy at the University of Aberdeen. He owns approximately 2.5 million LEGO bricks, teaches courses on comics, and has cats named Freckles and Mr. Prickely. He is the author of The Yablo Paradox: An Essay on Circularity (OUP, 2014). You can view Roy's other blog posts via his column on the OUPblog.

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Recent Comments

I don’t think “I am never” counts as a grammatical sentence. The only way to remove a word is to turn it into “I am periphrastic”. Now let’s assume that our initial sentence is true. Then it isn’t periphrastic, but we can remove a word to make a sentence that is true, namely “I am periphrastic”, which is periphrastic. So our initial sentence is false.

OK, so assume the initial sentence is false. Remove a word, and you get a sentence that’s true. So the initial sentence is not periphrastic, which is (more or less) what it states, so it’s true.

Roy T Cook24th September 2016

Steve,

I think there is a problem with your reasoning – you have assumed that “I am periphrastic” is periphrastic. But as I show (I think!) in the original post, this sentence might be periphrastic, but it might not.

This is a very interesting puzzle. Will [7] ‘I am unperiphrastic’ work as a paradox related to your periphrastic hypodox? Being unperiphrastic, a sentence would be such that either there is no way of removing one word to have a sensible sentence or there is a way of removing one word such that (every way where) a resulting sentence (makes sense, it) has the opposite truth value. Grant that [2] ‘I am’ is code for ‘This sentence exists’, and that these are true sentences, as I believe these are granted to formulate your periphrastic hypodox [3] (being a consistent conundrum that is underdetermined). Obviously [7] maps to [2] by removing one word (and [2] is the only sensible result of removing one word from [7]). Assume [7] is periphrastic. By the definition of periphrastic then [7] is true like [2]; hence, it is true that [7] is unperiphrastic, contradicting the assumption. Therefore, [7] is not periphrastic, but perforce of not being periphrastic, then it is unperiphrastic. But that is what [7] says, so [7] is true, in which case it has the same truth value as its derivative [2], and is periphrastic. Contradiction. Thus, the semantic value of [7] is overdetermined, and hence it is paradoxical. I hope this is all right.As one who conjectures that every hypodox has a related paradox, this is of particular interest to me. Do you agree [7] is a related paradox? best regards, Peter E-S